A coin is tossed indefinitely. Assume that the tosses are independent and let p be the probability of heads in each toss. For each k = 0,1.. let Ak be the event that k consecutive heads appear (in the 2^k tosses)
between the 2^k th toss(included) and 2^(k+1)st toss(not included). Show that if p >= 1/2 , then infinitely many of the events Ak occur with probability 1.

And the second almost same:

A coin is tossed indefinitely. Assume that the tosses are independent and let p be the probability of heads in each toss. For each k = 0,1.. let Ak be the event that k consecutive heads appear (in the 2^k tosses)
between the 2^k th toss(included) and 2^(k+1)st toss(not included). Show that if p < 1/2 , then infinitely many of the events Ak occur with probability 1.